Profile decomposition and multiple positive solutions for the perturbed CR Yamabe equation on the Heisenberg group

Abstract: In this article, we study an inhomogeneous critical nonlinear equation involving the sub-Laplacian on the Heisenberg group $\mathbb H<sup>n$. We prove the multiplicity of positive solutions for the critical problem \begin{align*} \mathcal{L}{\mathbb H^n} u=|u|^{2^\star-2}u+f(ξ) \quad \text{in } \mathbb H^n, \qquad u>0,\quad u\in S^{1,2}(\mathbb H^n), \end{align*} where $\mathcal{L}</em>{\mathbb H^n}$ is the sub-Laplacian on $\mathbb H<sup>n$, $2<sup>\star=\frac{2Q}{Q-2}$, $Q=2n+2$, $n\geq 1$, $S^{{1,2}(\mathbb} H<sup>n)$ is the homogeneous Sobolev space on $\mathbb H<sup>n$, and $f$ is a nontrivial nonnegative functional in the dual space $(S^{{1,2}(\mathbb} H<sup>n))&#39;$ satisfying a suitable smallness condition. The above mentioned equation appeared as a perturbation of the CR Yamabe equation on the Heisenberg group. A major difficulty comes from the lack of compactness of the critical Folland-Stein embedding into critical Lebesgue space. To overcome this, we establish a Palais-Smale profile decomposition for the associated energy functional. The obtained Palais-Smale profile decomposition identifies the precise energy levels at which lack of compactness may occur via energy quantization, and shows that every noncompact Palais-Smale sequence decomposes into a finite superposition of weakly interacting bubbles. As a key analytic ingredient, we establish an improved Folland-Stein-Sobolev inequality involving the Morrey norm, which serves as a fundamental interpolation inequality and plays a crucial role in detecting the concentration of noncompact Palais-Smale sequences.

• The paper establishes a complete Palais–Smale profile decomposition for the perturbed critical equation, showing that noncompact sequences split into a weak solution and finitely many asymptotically orthogonal Jerison–Lee bubbles with exact energy decoupling.
• It introduces a Morrey–Besov improved Folland–Stein inequality to detect concentration on the full Heisenberg group and proves that each extracted bubble removes a definite amount of energy, ensuring finite decomposition.
• For constant coefficient a≡1, the paper proves two distinct positive solutions when ||f||_(S¹)' < C₀S_Q^(Q/4), using variational minimization and a mountain-pass argument below the single-bubble energy threshold.

Setting and problem

This paper, by Basak, Chakraborty, Rana, and Roychowdhury (2608.16352), studies the inhomogeneous critical subelliptic equation on the Heisenberg group $\mathbb{H}^n$:

$$\mathcal{L}_{\mathbb{H}^n}u = a(\xi)|u|^{2^\star-2}u + f(\xi), \qquad u>0,\; u\in S^{1}(\mathbb{H}^n),$$

where $Q=2n+2$ is the homogeneous dimension and $2^\star = 2Q/(Q-2)$ the critical Folland–Stein exponent. The coefficient $a$ satisfies Assumption (A): $a\in L^\infty\cap C$, bounded below by a positive constant, with $a(\xi)\to 1$ at infinity. The perturbation $f\not\equiv 0$ is a nonnegative functional in $(S^1)'$. When $a\equiv 1$, $$\mathcal{L}_{\mathbb{H}^n}u = a(\xi)|u|^{2^\star-2}u + f(\xi), \qquad u>0,\; u\in S^{1}(\mathbb{H}^n),$$0, the equation reduces to the CR Yamabe equation, whose positive solutions are completely classified by the Jerison–Lee bubbles $$\mathcal{L}_{\mathbb{H}^n}u = a(\xi)|u|^{2^\star-2}u + f(\xi), \qquad u>0,\; u\in S^{1}(\mathbb{H}^n),$$1 [JL88]. The inhomogeneous term breaks the scale and translation invariances of $$\mathcal{L}_{\mathbb{H}^n}u = a(\xi)|u|^{2^\star-2}u + f(\xi), \qquad u>0,\; u\in S^{1}(\mathbb{H}^n),$$2, so new solutions beyond the bubble family become possible.

The central obstruction is the failure of compactness of the critical Folland–Stein embedding $$\mathcal{L}_{\mathbb{H}^n}u = a(\xi)|u|^{2^\star-2}u + f(\xi), \qquad u>0,\; u\in S^{1}(\mathbb{H}^n),$$3, driven by left-translation and non-isotropic dilation invariance. The paper's two main contributions are: (i) a complete Palais–Smale profile decomposition for the energy functional $$\mathcal{L}_{\mathbb{H}^n}u = a(\xi)|u|^{2^\star-2}u + f(\xi), \qquad u>0,\; u\in S^{1}(\mathbb{H}^n),$$4 on the full Heisenberg group — previously available only on bounded domains via Palatucci–Piccinini–Temperini [PPT25] and not at all for the nonhomogeneous problem — and (ii) a multiplicity theorem yielding two distinct positive solutions when $$\mathcal{L}_{\mathbb{H}^n}u = a(\xi)|u|^{2^\star-2}u + f(\xi), \qquad u>0,\; u\in S^{1}(\mathbb{H}^n),$$5 is small.

Improved Folland–Stein–Sobolev inequality

The concentration-detection mechanism replaces Lions' concentration-compactness principle with an interpolation inequality involving Morrey norms. The key new ingredient is an embedding of Morrey spaces into negative-order Besov spaces on $$\mathcal{L}_{\mathbb{H}^n}u = a(\xi)|u|^{2^\star-2}u + f(\xi), \qquad u>0,\; u\in S^{1}(\mathbb{H}^n),$$6: for $$\mathcal{L}_{\mathbb{H}^n}u = a(\xi)|u|^{2^\star-2}u + f(\xi), \qquad u>0,\; u\in S^{1}(\mathbb{H}^n),$$7,

$$\mathcal{L}_{\mathbb{H}^n}u = a(\xi)|u|^{2^\star-2}u + f(\xi), \qquad u>0,\; u\in S^{1}(\mathbb{H}^n),$$8

proved by dyadic annular decomposition combined with Gaussian upper bounds on the heat kernel. Coupled with Chamorro's improved Sobolev inequality in Besov spaces [C11], this yields: for $$\mathcal{L}_{\mathbb{H}^n}u = a(\xi)|u|^{2^\star-2}u + f(\xi), \qquad u>0,\; u\in S^{1}(\mathbb{H}^n),$$9, $Q=2n+2$0, and $Q=2n+2$1,

$Q=2n+2$2

The authors note that since both constituent results hold on general stratified Lie groups, the inequality extends verbatim beyond $Q=2n+2$3. For $Q=2n+2$4, the inequality forces any bounded sequence that does not vanish in $Q=2n+2$5 to have Morrey norm uniformly bounded below, which localizes concentration to a specific Korányi ball.

Palais–Smale profile decomposition

The first main result describes precisely how compactness fails. For any $Q=2n+2$6 sequence of $Q=2n+2$7 under (A) and (F), there exist a weak solution $Q=2n+2$8 of the perturbed problem and finitely many profiles $Q=2n+2$9 solving limiting equations $2^\star = 2Q/(Q-2)$0, where $2^\star = 2Q/(Q-2)$1 along some non-identity point $2^\star = 2Q/(Q-2)$2. The sequence decomposes as

$2^\star = 2Q/(Q-2)$3

with asymptotic orthogonality of distinct bubbles,

$2^\star = 2Q/(Q-2)$4

and exact energy decoupling:

$2^\star = 2Q/(Q-2)$5

Three technical components deserve emphasis. First, Lemma on orthogonality characterizes exactly when two families of dilated translations become asymptotically orthogonal in $2^\star = 2Q/(Q-2)$6, the Heisenberg analogue of Gérard's Euclidean characterization [Ger98]; its proof handles both diverging scales ($2^\star = 2Q/(Q-2)$7) and concentrating scales ($2^\star = 2Q/(Q-2)$8) separately. Second, a weighted Brezis–Lieb lemma for the critical nonlinearity with $2^\star = 2Q/(Q-2)$9 is established via the standard algebraic inequality and dominated convergence. Third, each extracted profile is upgraded from weak nonnegative solution to strictly positive classical solution using subelliptic regularity and the weak Harnack inequality, then identified as a Jerison–Lee bubble by the classification theorem. Each iteration drops the energy by at least $a$0, so the iteration terminates after finitely many steps.

An immediate corollary: if $a$1 holds at level $a$2, then $a$3 cannot equal $a$4 for any $a$5.

Multiplicity of positive solutions

For the constant-coefficient problem ($a$6), the paper proves that if

$a$7

then the problem admits at least two distinct positive solutions. The proof uses a fiber-map partition of $a$8 via $a$9 into sets homeomorphic to the ball, sphere, and exterior region, following the Euclidean strategy of Bhakta–Pucci [BP20].

First solution: a local minimizer $a\in L^\infty\cap C$0 of $a\in L^\infty\cap C$1 in $a\in L^\infty\cap C$2 with negative energy $a\in L^\infty\cap C$3, obtained via Ekeland's variational principle. Compactness follows because any bubble would force $a\in L^\infty\cap C$4 while pushing $a\in L^\infty\cap C$5, contradicting the strict inequality $a\in L^\infty\cap C$6 established through a careful one-dimensional analysis of the ray functional.

Second solution: a mountain-pass critical point at level $a\in L^\infty\cap C$7 satisfying the sharp two-sided bound

$a\in L^\infty\cap C$8

Since each Jerison–Lee bubble carries energy exactly $a\in L^\infty\cap C$9, the upper bound categorically excludes bubbling ($a(\xi)\to 1$0), forcing strong convergence of the $a(\xi)\to 1$1 sequence. This is where the explicit extremal classification and sharp constant $a(\xi)\to 1$2 are indispensable: without them, the quantized energy drop cannot be computed and the threshold argument collapses.

Positivity of both critical points follows from testing against $a(\xi)\to 1$3 (using nonnegativity of $a(\xi)\to 1$4) together with the subelliptic strong maximum principle, which propagates strict positivity via horizontal path connectivity.

Limitations and open questions

The authors state plainly that the framework does not currently extend to fractional sub-Laplacians or general homogeneous Lie groups: the precise energy quantization relies fundamentally on the explicit Jerison–Lee classification and the sharp Sobolev constant $a(\xi)\to 1$5, neither of which is known in those settings. The multiplicity result is proved only for $a(\xi)\to 1$6; the variable-coefficient case is covered only at the level of the profile decomposition, and extending the two-solution theorem to general $a(\xi)\to 1$7 satisfying (A) remains open. The smallness condition on $a(\xi)\to 1$8 is quantitative but its optimality is not addressed. Finally, whether the number of solutions can be sharpened or related to topological data, as in the Bahri–Coron theory on bounded domains, is not investigated here.

Conclusion

The paper provides the first complete Palais–Smale profile decomposition on the full Heisenberg group for a nonhomogeneous critical problem, built on a new Morrey–Besov improved Folland–Stein inequality, and converts it into a concrete multiplicity theorem: two positive solutions exist whenever the dual norm of the perturbation falls below the explicit threshold $a(\xi)\to 1$9. The argument demonstrates how the rigidity of the homogeneous CR Yamabe equation — complete classification of extremals — becomes a quantitative tool for excluding loss of compactness in the perturbed regime.